Is there a nice explicit description for the group $K^*/Nm_{L/K}(L^*)$ for a finite field extension $L/K$?

What if for example, $L$ is obtained from $K$ by ajoining an n-th root of some $\alpha \in K$ (and assuming that $K$ contains the n-th root of unity)? I don't see a nice answer even for the case $n=2$.

Thank you.

EDIT: Thanks for the answers! Is that correct that in the case of a cyclic extension this group is isomorphic to $Br(L/K)$, since both these groups are identified with $H^2(Gal(L/K), L^*)$?